I currently have Spiceworks set up to run as a serivce, and as such, it no longer shows up in the System Tray. I received notification of the new release to Spice Works and would like to update my install. How would I go about doing this?

That said, at any time you can always run a manual update by going to spiceworks.com\download & getting the latest version. I typically recommend backing up first, but I have yet to experience any issues during the upgrade process.

You can just go to the server and log in as an admin, then click on the spiceworks desktop icon on your start menu, if it its missing go to "C:\Program Files\Spiceworks\spiceworks_desktop.exe" that will launch the desktop tray manager.
Alternatively you can just download the update and install it on the server the update will recognize the installation and update the software correctly